ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The aim of the paper is to prove the following result: Given any finitely generated algebra $A$ of finite similarity type, there exists a $2$-unary algebra $B$ such that $B$ is generated by any of its elements and ${\rm Con}B$ is isomorphic to ${\rm Con}A$.

AMS Subject Classification
(1991): 08A30

Received January 9, 1997 and in revised form August 5, 1997. (Registered under 2641/2009.)

Abstract. The paper investigates the degree of categorical non-universality of the class of monounary algebras. It characterizes algebras whose endomorphisms are invertible, describes all abelian automorphism groups of monounary algebras and then represents each such group as the automorphism group by a proper class of pairwise non-isomorphic monounary algebras, and lists the isomorphism types of their commutative endomorphism monoids.

Abstract. Let $F_{p,t} (n)$ denote the number of the coefficients of $(x_1 + x_2 +\cdots + x_t)^j$, $0\le j\le n-1$, which are not divisible by the prime $p$. Then we have $\alpha(p,t) = \limsup F_{p,t} (n)/n^\theta =1$, and $\beta(p,t)=\liminf F_{p,t} (n) /n^\theta $ can be calculated to a given precision, where $\theta = \log{p+t-1\choose t} \big/ \log p$.

AMS Subject Classification
(1991): 11A63, 11K16

Received October 28, 1997 and in revised form January 26, 1998. (Registered under 2643/2009.)

Abstract. It is proved that if $f,g\colon{\msbm N}\to\{0,1\} $ are completely multiplicative functions such that $g(an+b)=f(n)$ is satisfied for some positive coprime integers $a$, $b$ and for every positive integer $n$, furthermore $g(p)=0$ for all primes $p| b$, then $f(n)=1$ iff $(n,b)=1$ and $g(n)=1$ for $(n,ab)=1$, $g(n)=0$ for $(n,b)>1$.

AMS Subject Classification
(1991): 11N64, 11N69

Received November 18, 1996 and in revised form November 5, 1997. (Registered under 2644/2009.)

Abstract. The theorem we prove in this paper provides a unified and elementary method to prove various density theorems independent of the wellknown Lebesgue density theorem, but including density theorems like the hearts density theorem by Aversa and Preiss.

AMS Subject Classification
(1991): 26A24, 26A99

Received March 3, 1997 and in revised form September 18, 1997. (Registered under 2645/2009.)

Abstract. A function $\varphi $ is called T-universal on an open set ${\cal O} \subset{\msbm C}$ if $\varphi $ is holomorphic on ${\cal O}$ and satisfies the following properties. For all compact sets $K$ with connected complement, for all functions $f$ which are continuous on $K$ and holomorphic in its interior and for all $\zeta\in \partial{\cal O}$ there exists a sequence $\{(a_n, b_n)\} $ in ${\msbm C}^2$ with $a_n z + b_n \in{\cal O}$ for all $z \in K$ and all $n \in{\msbm N}$, such that $\{a_n z + b_n\} $ converges to $\zeta $ and $\{\varphi(a_n z + b_n)\} $ converges to $f (z)$ uniformly on $K$. The existence of T-universal functions is proved for open sets ${\cal O}$ with simply connected components. If ${\cal O}$ contains the origin, then $\varphi $ can be chosen with a lacunary power series $\varphi(z) = \sum ^\infty_{\nu = 0} \varphi_\nu z^\nu $, where $\varphi_\nu = 0$ for $\nu\not\in Q$ with a certain prescribed set $Q \subset{\msbm N}_0$.

AMS Subject Classification
(1991): 30B10, 30E10, 30B60

Received April 23, 1997 and in revised form September 4, 1997. (Registered under 2646/2009.)

Abstract. This paper is devoted to the problems of recapturing a Pick function from its radial boundary values and (estimates of) radial derivatives at points of a finite subset of ${\msbm R}$ and from (estimates of) its radial residues at points of another finite subset of ${\msbm R}$. Necessary and sufficient conditions for solvability of these problems are established, in terms of nonnegativity and rank of the Pick matrix.

AMS Subject Classification
(1991): 30E05, 46E22

Received April 2, 1997 and in revised form June 18, 1997. (Registered under 2647/2009.)

Abstract. Szegő's theorem gives an explicit expression for the infinum of weighted $L^p$-norms over the circle for normalized analytic functions on the disc. The main result of the paper gives, in terms of the metrics on the Smirnov class $N^+$ and the classes $N^p$ $(1< p< \infty )$, the logarithmic version of Szegő's theorem.

Abstract. We propose a new method to prove the strong asymptotic stability of isotropic elasticity systems with internal damping. Unlike the earlier works, our method also applies in the case of feedbacks with no growth assumptions at the origin.

Abstract. In this paper, we study the observability, the controllability and the boundary stabilizability of the linear elasticity systems. This work extends to non-isotropic systems with variable coefficients, the observability and exact controllability results for isotropic elastodynamic systems obtained by J.-L. Lions in 1988, the uniform stabilizability results for two-dimensional isotropic systems obtained by J. E. Lagnese in 1991 and the results obtained by Alabau and Komornik [1].

Abstract. We consider homogenization of sequences of integral functionals with natural growth conditions. Some new bounds for the corresponding homogenized integrand are proved and discussed. These bounds are compared with the nonlinear bounds of Wiener and Hashin--Shtrikman type. We also point out conditions that make our bounds sharp.

Abstract. We prove a Tauberian theorem concerning the summability method $D_{\lambda,a}$ based on the Dirichlet series $\sum a_ne^{-\lambda_nx}$ with $a_{n+1}\sim a_n>0$ when the sequence $(\lambda_n)$ satisfies the `high indices' condition $\lambda_{n+1}>c\lambda_n\ge0$ with any $c>1.$

Abstract. In this paper we investigate linear operators between certain sequence spaces $X$ and $Y$. Among other things, if $X$ is any $p$--normed space and $Y = w_0^1, w^1, w_{\infty }^1, c_0(\mu ), c(\mu )$, or $c_{\infty }(\mu )$ we find necessary and sufficient conditions for $A$ to map $X$ into $Y$. Then the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for $A$ to be a compact operator.

Abstract. Alexits' theorem is generalized to the case of $L^\infty $ space on the real line. Known results were obtained under additional restrictive assumptions. We avoid these by introducing a new analog of conjugate function.

Abstract. We consider sublinear operators defined on (dyadic) martingales and give sufficient conditions for them to be bounded from the dyadic Hardy space $ H^p $ to itself. Especially, multiplier operators and their transforms will be considered from $ H^p $ to $ H^p $ for some "powers" $ p $. As a consequence we obtain that for these $ p $'s the space $ H^p $ can be characterized by means of the so-called Sunouchi operator $U$. Namely, a martingale $f$ belongs to $H^p$ ($1/2< p\leq1$, $Sf=0$) if and only if $Uf\in L^p $. Furthermore, $\|f\|_{H^p}\sim\|Uf\|_p $.

AMS Subject Classification
(1991): 42C10, 42B15, 43A75, 60G42

Received April 24, 1997 and in revised form September 1, 1997. (Registered under 2655/2009.)

Abstract. Our goal is to show that the dyadic Cesàro operator is bounded on $L^p[0,1)$ $(1\le p< \infty )$ and on the dyadic Hardy space $H^1[0,1)$ and isn't bounded on the spaces VMO and on $L^\infty[0,1)$. Due to the duality we can easily discuss the boundedness of the Copson operator, which is the adjoint operator of the Cesàro operator. The boundedness of the Cesàro operator on $L^p$ $(1\le p< \infty )$ and on the Hardy space has been already examined for the trigonometric system (see [6] and [3]).

AMS Subject Classification
(1991): 42C10, 42B30

Received May 20, 1997 and in revised form September 23, 1997. (Registered under 2656/2009.)

Abstract. Asymptotics is obtained for the Lebesgue constants of the Cesàro means of spherical harmonic expansions. Precise constant in the main term is found and the order of growth of the remainder term is given.

AMS Subject Classification
(1991): 42C10, 43A90, 33D55, 442B08

Received March 17, 1997 and in revised form June 9, 1997. (Registered under 2657/2009.)

Abstract. It is shown that given a co-hyponormal and quasitriangular operator $T$ with connected spectrum, there exists a compact $K$ such that $T+K$ is strongly irreducible. Using the above result, we prove that every analytic {\it Toeplitz operator} on Bergman space $L_a^2({\cal B}_{n}, dv)$ is the sum of a strongly irreducible Toeplitz operator and a Berezin perturbation, where ${\cal B}_{n}$ is the unite ball of complex n-dimensional space and $n\geq1$.

Abstract. Given a $p$-hyponormal operator $(0< p< 1/2)$ $T$ on a Hilbert space, define operators $\hat T$ and $\tilde T$ by $\hat T=| T| ^{1/2}U| T| ^{1/2}$ and $\tilde T=| \hat T| ^{1/2}V| \hat T| ^{1/2}$, where the partial isometries $U$ and $V$ are as in the polar decompositions $T=U| T| $ and $\hat T=V| \hat T| $. The operator $\tilde T$ is then hyponormal. We show that $T$ has a non-trivial invariant subspace if and only if $\tilde T$ does, and that if $T$ does not have a non-trivial invariant subspace, then $T$ is the compact perturbation of a normal operator. We also consider upper triangular operators with $p$-hyponormal entries along the main diagonal.

Abstract. We consider various Weyl's theorems in connection with the continuity of the reduced minimum modulus, Weyl spectrum, Browder spectrum, essential approximate point spectrum and Browder essential approximate point spectrum. If $H$ is a Hilbert space, and $T\in B(H)$ is a quasihyponormal operator, we prove the spectral mapping theorem for the essential approximate point spectrum and for arbitrary analytic function, defined on some neighbourhood of $\sigma(T)$. Also, if $T^*$ is quasihyponormal, we prove that the $a$-Weyl's theorem holds for $T$.

Abstract. We consider the Riemann means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)} ({\msbm T}^2)$, $H^{(0,1)} ({\msbm T}^2)$, or $H^{(1,1)} ({\msbm T}^2)$, where ${\msbm T}^2 := [-\pi, \pi ) \times[-\pi, \pi )$ is the two-dimensional torus. We prove that the maximal Riemann operator is bounded from $H^{(1,0)} ({\msbm T}^2)$ or $H^{(0,1)} ({\msbm T}^2)$ into weak-$L^1({\msbm T}^2)$, and conjecture that it is bounded from $H^{(1,1)}({\msbm T}^2)$ into $L^1({\msbm T}^2)$. As corollaries, we obtain analogous results on the maximal conjugate Riemann operators, as well as we deduce the pointwise convergence of both the Riemann and the conjugate Riemann means almost everywhere.

AMS Subject Classification
(1991): 47B38, 42A50, 42B08

Received March 24, 1997 and in revised form December 3, 1997. (Registered under 2663/2009.)

Abstract. This is a second part of a paper that makes a continuation of our paper "Equivalence of shape fibrations and approximate fibrations" which also explores further some basic properties of shape and approximate fibrations of arbitrary topological spaces using relations. Our method is again to use relations with smaller and smaller images of points. The paper is self-reliant and does not require extensive knowledge of relations.